\begin{block}{Sample coefficient of correlation} Assume that we have $n$ measurements on two variables $x$ and $y$.\\
Denote the mean and the standard deviation of the variable $x$ with $\bar{x}$ and $s_{x}$ and the mean and the standard deviation of the $y$ variable with $\bar{y}$ and $s_{y}$. \\
The sample coefficient of correlation is \[ r = \frac{1}{n-1} \sum_{i=1}^n \left( \frac{x_i - \bar{x}}{s_x}\right) \left( \frac{y_i - \bar{y}}{s_y}\right). \] \end{block}
{\bf Warning:} The correlation only estimates the strength of a {\bf linear} relationship!
Details
\begin{block}{Sample coefficient of correlation} Assume that we have $n$ measurements on two variables $x$ and $y$.\\
Denote the mean and the standard deviation of the variable $x$ with $\bar{x}$ and $s_{x}$ and the mean and the standard deviation of the $y$ variable with $\bar{y}$ and $s_{y}$. \\
The sample coefficient of correlation is \[ r = \frac{1}{n-1} \sum_{i=1}^n \left( \frac{x_i - \bar{x}}{s_x}\right) \left( \frac{y_i - \bar{y}}{s_y}\right). \] \end{block}
{\bf Warning:} The correlation only estimates the strength of a {\bf linear} relationship!